from ∞Grpd≃Sh∞(*)\simeq Sh_\infty(\ast) to 𝒳\mathcal{X}. We say that 𝒳\mathcal{X} has enough points if a 1-morphismf:X→Yf \colon X \to Y in 𝒳\mathcal{X} in an equivalence in 𝒳\mathcal{X} precisely if for all such points the inverse imagep*(f)p^\ast (f) (the stalk at the point) is an equivalence in ∞Grpd.

Remark

There exist 1-sitesCC such that the (1,1)-topos of sheaves of sets on CC has enough points in the 1-topos sense, but such that the corresponding 1-localic (∞,1)-topos𝒳\mathcal{X} does not have enough points in the sense of def. 2. An example is given by the site of open subsets of the topological space ∏ℕ{x,z,y}\prod_{\mathbb{N}} \{x,z,y\} where the topology on {x,z,y}\{x,z,y\} is generated by the two open subsets {x,y}\{x,y\} and {x,z}\{x,z\}. See HTT, Remark 6.5.4.7.

Proof

The strategy is to form the localization in a 2-step process, where we first just form the Cech-localization, and then from that the full hypercompletion. For that notice that among the weak equivalences in the Joyal-Jardine local model structure on simplicial presheaves are in particular the ordinary covering sieves S({Ui})↪j(U)S(\{U_i\}) \hookrightarrow j(U) (here jj is the Yoneda embedding) associated with a coverin family {Ui→U}\{U_i \to U\} in the siteCC:

since CC is an ordinary category, the simplicial presheaves S({Ui})S(\{U_i\}) and j(U)j(U) have vanishing presheaves of homotopy groups in positive degree, while they coindide with their π0\pi_0-presheaves. Since the sheafification of S({Ui})S(\{U_i\}) is isomorphic to j(U)j(U), by definition, it follows that the same holds for the π0\pi_0-presheaves and trivially for the πn\pi_n-presheaves. So S({Ui})→j(U)S(\{U_i\}) \to j(U) is a Joyal-Jardine weak equivalence.

We will now first localize with respect to these morphisms to obtain the Cech-localization whose fibrant objects are (∞,1)-sheaves. The point is that on these fibrant objects then, the Joyal-Jardine sheaves of homotopy groups can be seen to coincide with the (∞,1)-categorical homotopy sheaves in terms of which hypercompletion is defined.

By the above remark, the Joyal-Jardine localization [Cop,sSet]proj,loc[C^{op}, sSet]_{proj,loc} that we are after is a further localization of this Cech localization : we have the bottom row in the following diagram, and want to see that the top left corner is as indicated:

Furthermore, if X∈[Cop,sSet]inj,covX \in [C^{op}, sSet]_{inj,cov} is fibrant, it satisfies descent for simplicial presheaves at Cech covers. Since powering is a Quillen bifunctor, the same is then true for XSX^S, formed in the model category, so XSX^S is an ∞\infty-stack. But that means its 0-truncationτ≤0(XSn)\tau_{\leq 0}(X^{S^n}) is an ordinary sheaf. (Observe that truncation commutes with localization, as discussed here.)

In total this shows that on fibrant objectsXX in [Cop,sSet]inj,cov[C^{op}, sSet]_{inj,cov}, the Joyal-Jardine homotopy sheaves coincide with the (∞,1)(\infty,1)-categorical homotopy sheaves of the object XX.

It remains to observe that under left Bousfield lcoalization, the new fibrant objects are precisely those old fibrant objects that are also local objects with respect to the morphisms at which one localizes. With the above this implies that the left Bousfield localization [Cop,sSet]inj,cov→[Cop,sSet]inj,loc[C^{op}, sSet]_{inj,cov} \to [C^{op}, sSet]_{inj,loc} does model the hypercompletionSh(∞,1)(C)→Sh(∞,1)^(C)Sh_{(\infty,1)}(C) \to \widehat {Sh_{(\infty,1)}}(C).

In classical topos theory

In classical topos theory literature frequently simplicial objects in an ordinary topos are considered, with acyclic fibrations taken to be those morphisms Y•→X•Y_\bullet \to X_\bullet such that for all horn inclusions the induced morphism